Numerische Mathematik

, Volume 105, Issue 3, pp 413–455 | Cite as

The tent transformation can improve the convergence rate of quasi-Monte Carlo algorithms using digital nets

  • Ligia L. Cristea
  • Josef Dick
  • Gunther Leobacher
  • Friedrich Pillichshammer
Article

Abstract

In this paper we investigate multivariate integration in reproducing kernel Sobolev spaces for which the second partial derivatives are square integrable. As quadrature points for our quasi-Monte Carlo algorithm we use digital (t,m,s)-nets over \(\mathbb{Z}_2\) which are randomly digitally shifted and then folded using the tent transformation. For this QMC algorithm we show that the root mean square worst-case error converges with order \(2^{m(-2+\varepsilon)}\) for any ɛ >  0, where 2m is the number of points. A similar result for lattice rules has previously been shown by Hickernell.

Mathematics Subject Classification (1991)

11K38–11K06 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Ligia L. Cristea
    • 1
  • Josef Dick
    • 2
  • Gunther Leobacher
    • 1
  • Friedrich Pillichshammer
    • 1
  1. 1.Institut für FinanzmathematikUniversität LinzLinzAustria
  2. 2.Division of Engineering, Science & TechnologyUNSW AsiaSingaporeSingapore

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